A curve \(C\) has equation \(y=ax+b+\frac{2b-a}{x+3}\), where \(a\) and \(b\) are constants such that \(a>0\), \(b e \frac{1}{2}a\) and \(x e -3\).
If \(C\) has no stationary points, use differentiation to find the relationship between \(a\) and \(b\).[3]
It is now given that \(b=a\).
Find the \(x\)-coordinates of the turning points.[2]
Hence sketch \(C\), stating the equations of any asymptotes and the coordinates of the axial intercepts and turning points.[3]
Verify that the point of intersection of the two asymptotes of \(C\) lies on the line \(y=kx+3k-2a\). Hence, using the graph in part (b)(ii), find the range of values of \(k\) in terms of \(a\), such that the equation \(x(k-a)+3k-2a-b-\frac{2b-a}{x+3}=0\) has real roots.[2]